Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [87,4,Mod(4,87)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("87.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 87 = 3 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 87.i (of order \(14\), degree \(6\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(5.13316617050\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −5.32561 | − | 1.21553i | −1.30165 | − | 2.70291i | 19.6768 | + | 9.47585i | 1.36680 | − | 5.98834i | 3.64660 | + | 15.9768i | −25.4953 | + | 12.2779i | −59.1063 | − | 47.1357i | −5.61141 | + | 7.03648i | −14.5581 | + | 30.2302i |
4.2 | −5.09551 | − | 1.16302i | 1.30165 | + | 2.70291i | 17.4039 | + | 8.38126i | −2.72939 | + | 11.9583i | −3.48905 | − | 15.2865i | 11.6156 | − | 5.59376i | −46.2437 | − | 36.8782i | −5.61141 | + | 7.03648i | 27.8153 | − | 57.7591i |
4.3 | −3.92262 | − | 0.895312i | −1.30165 | − | 2.70291i | 7.37761 | + | 3.55287i | −1.96973 | + | 8.62997i | 2.68594 | + | 11.7679i | 21.0311 | − | 10.1281i | −0.593065 | − | 0.472954i | −5.61141 | + | 7.03648i | 15.4530 | − | 32.0886i |
4.4 | −3.58771 | − | 0.818871i | 1.30165 | + | 2.70291i | 4.99335 | + | 2.40467i | 1.45209 | − | 6.36202i | −2.45661 | − | 10.7631i | −16.7086 | + | 8.04642i | 7.07137 | + | 5.63923i | −5.61141 | + | 7.03648i | −10.4193 | + | 21.6360i |
4.5 | −2.51168 | − | 0.573274i | −1.30165 | − | 2.70291i | −1.22787 | − | 0.591312i | 4.17965 | − | 18.3122i | 1.71982 | + | 7.53503i | −3.13542 | + | 1.50994i | 18.8587 | + | 15.0393i | −5.61141 | + | 7.03648i | −20.9958 | + | 43.5983i |
4.6 | −2.42711 | − | 0.553971i | 1.30165 | + | 2.70291i | −1.62379 | − | 0.781976i | 0.563333 | − | 2.46813i | −1.66191 | − | 7.28132i | 6.69997 | − | 3.22654i | 19.0790 | + | 15.2150i | −5.61141 | + | 7.03648i | −2.73454 | + | 5.67833i |
4.7 | −1.72017 | − | 0.392617i | −1.30165 | − | 2.70291i | −4.40292 | − | 2.12033i | −1.38401 | + | 6.06376i | 1.17785 | + | 5.16050i | −7.07166 | + | 3.40553i | 17.7770 | + | 14.1767i | −5.61141 | + | 7.03648i | 4.76148 | − | 9.88731i |
4.8 | 0.0265736 | + | 0.00606526i | 1.30165 | + | 2.70291i | −7.20708 | − | 3.47075i | −0.0451194 | + | 0.197681i | 0.0181958 | + | 0.0797209i | −15.3287 | + | 7.38191i | −0.340951 | − | 0.271899i | −5.61141 | + | 7.03648i | −0.00239797 | + | 0.00497944i |
4.9 | 1.30198 | + | 0.297169i | −1.30165 | − | 2.70291i | −5.60090 | − | 2.69725i | −4.08818 | + | 17.9115i | −0.891506 | − | 3.90594i | 1.77437 | − | 0.854493i | −14.8436 | − | 11.8374i | −5.61141 | + | 7.03648i | −10.6455 | + | 22.1056i |
4.10 | 1.49974 | + | 0.342306i | 1.30165 | + | 2.70291i | −5.07570 | − | 2.44433i | 4.74328 | − | 20.7817i | 1.02692 | + | 4.49922i | 10.9198 | − | 5.25869i | −16.3971 | − | 13.0763i | −5.61141 | + | 7.03648i | 14.2274 | − | 29.5434i |
4.11 | 1.76854 | + | 0.403657i | −1.30165 | − | 2.70291i | −4.24297 | − | 2.04330i | 1.80833 | − | 7.92279i | −1.21097 | − | 5.30561i | −26.4428 | + | 12.7342i | −18.0251 | − | 14.3745i | −5.61141 | + | 7.03648i | 6.39618 | − | 13.2818i |
4.12 | 2.00965 | + | 0.458689i | 1.30165 | + | 2.70291i | −3.37946 | − | 1.62746i | −3.83335 | + | 16.7950i | 1.37607 | + | 6.02895i | −24.5544 | + | 11.8248i | −18.9379 | − | 15.1025i | −5.61141 | + | 7.03648i | −15.4074 | + | 31.9937i |
4.13 | 2.73476 | + | 0.624190i | −1.30165 | − | 2.70291i | −0.118474 | − | 0.0570538i | 0.858190 | − | 3.75998i | −1.87257 | − | 8.20427i | 27.5583 | − | 13.2714i | −17.8332 | − | 14.2215i | −5.61141 | + | 7.03648i | 4.69388 | − | 9.74694i |
4.14 | 3.86408 | + | 0.881950i | 1.30165 | + | 2.70291i | 6.94550 | + | 3.34478i | −2.23984 | + | 9.81340i | 2.64585 | + | 11.5922i | 26.9760 | − | 12.9910i | −0.901956 | − | 0.719286i | −5.61141 | + | 7.03648i | −17.3099 | + | 35.9443i |
4.15 | 4.81413 | + | 1.09879i | 1.30165 | + | 2.70291i | 14.7607 | + | 7.10840i | 1.38037 | − | 6.04778i | 3.29638 | + | 14.4424i | −11.0304 | + | 5.31195i | 32.3644 | + | 25.8098i | −5.61141 | + | 7.03648i | 13.2905 | − | 27.5981i |
4.16 | 4.87893 | + | 1.11358i | −1.30165 | − | 2.70291i | 15.3561 | + | 7.39513i | 3.82231 | − | 16.7467i | −3.34075 | − | 14.6368i | 0.370677 | − | 0.178508i | 35.3857 | + | 28.2191i | −5.61141 | + | 7.03648i | 37.2976 | − | 77.4493i |
13.1 | −2.41931 | + | 5.02376i | −2.34549 | − | 1.87047i | −14.3971 | − | 18.0534i | −5.35025 | − | 2.57654i | 15.0713 | − | 7.25794i | −10.9950 | + | 13.7872i | 82.0381 | − | 18.7247i | 2.00269 | + | 8.77435i | 25.8879 | − | 20.6449i |
13.2 | −2.22145 | + | 4.61288i | 2.34549 | + | 1.87047i | −11.3559 | − | 14.2399i | −10.0741 | − | 4.85142i | −13.8386 | + | 6.66434i | 21.3564 | − | 26.7801i | 50.9810 | − | 11.6361i | 2.00269 | + | 8.77435i | 44.7580 | − | 35.6933i |
13.3 | −1.93516 | + | 4.01839i | 2.34549 | + | 1.87047i | −7.41472 | − | 9.29777i | 11.6028 | + | 5.58764i | −12.0552 | + | 5.80547i | −13.8901 | + | 17.4176i | 16.9247 | − | 3.86294i | 2.00269 | + | 8.77435i | −44.9066 | + | 35.8118i |
13.4 | −1.78908 | + | 3.71506i | −2.34549 | − | 1.87047i | −5.61296 | − | 7.03842i | 16.3624 | + | 7.87972i | 11.1452 | − | 5.36724i | 21.8101 | − | 27.3490i | 4.03000 | − | 0.919821i | 2.00269 | + | 8.77435i | −58.5473 | + | 46.6899i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.e | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 87.4.i.a | ✓ | 96 |
29.e | even | 14 | 1 | inner | 87.4.i.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
87.4.i.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
87.4.i.a | ✓ | 96 | 29.e | even | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(87, [\chi])\).